Problem: Factor the following expression: $-5$ $x^2$ $-9$ $x+$ $18$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-5)}{(18)} &=& -90 \\ {a} + {b} &=& & & {-9} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-90$ and add them together. Remember, since $-90$ is negative, one of the factors must be negative. The factors that add up to ${-9}$ will be your ${a}$ and ${b}$ When ${a}$ is ${6}$ and ${b}$ is ${-15}$ $ \begin{eqnarray} {ab} &=& ({6})({-15}) &=& -90 \\ {a} + {b} &=& {6} + {-15} &=& -9 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-5}x^2 +{6}x {-15}x +{18} $ Group the terms so that there is a common factor in each group: $ ({-5}x^2 +{6}x) + ({-15}x +{18}) $ Factor out the common factors: $ x(-5x + 6) + 3(-5x + 6) $ Notice how $(-5x + 6)$ has become a common factor. Factor this out to find the answer. $(-5x + 6)(x + 3)$